<h2>Problem 105</h2>
<div style="color:#666;font-size:80%;">23 September 2005</div><br />
<div class="problem_content">
<p>Let S(A) represent the sum of elements in set A of size <i>n</i>. We shall call it a special sum set if for any two non-empty disjoint subsets, B and C, the following properties are true:</p>
<ol style="list-style-type:lower-roman;">
<li>S(B) <img src='images/symbol_ne.gif' width='11' height='10' alt='&ne;' border='0' style='vertical-align:middle;' /> S(C); that is, sums of subsets cannot be equal.</li>
<li>If B contains more elements than C then S(B) <img src='images/symbol_gt.gif' width='10' height='10' alt='&gt;' border='0' style='vertical-align:middle;' /> S(C).</li>
</ol>
<p>For example, {81, 88, 75, 42, 87, 84, 86, 65} is not a special sum set because 65 + 87 + 88 = 75 + 81 + 84, whereas {157, 150, 164, 119, 79, 159, 161, 139, 158} satisfies both rules for all possible subset pair combinations and S(A) = 1286.</p>
<p>Using <a href="project/sets.txt">sets.txt</a> (right click and "Save Link/Target As..."), a 4K text file with one-hundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, A<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" />, A<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />, ..., A<img src="" style="display:none;" alt="_(" /><sub><i>k</i></sub><img src="" style="display:none;" alt=")" />, and find the value of S(A<img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" />) + S(A<img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" />) + ... + S(A<img src="" style="display:none;" alt="_(" /><sub><i>k</i></sub><img src="" style="display:none;" alt=")" />).</p>
<p class="info">NOTE: This problem is related to problems <a href="index.php?section=problems&amp;id=103">103</a> and <a href="index.php?section=problems&amp;id=106">106</a>.</p>
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